#6653779

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632082
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#37277	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1494 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,6)(4,5)(7,8)(11,15)(14,16)(17,18,23,21)(19,24,22,30)(20,27,29,32)(25,31,28,26);(2,5)(3,7)(9,10)(11,14)(12,13)(15,16)(17,19,23,22)(18,24,21,30)(20,26,29,31)(25,32,28,27);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20)(18,25)(19,26)(21,28)(22,31)(23,29)(24,32)(27,30);(2,5)(6,8)(10,13)(14,16)(17,21,23,18)(19,24,22,30)(20,28,29,25)(26,32,31,27);(9,11,12,15)(10,14,13,16)(18,21)(24,30)(25,28)(27,32)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=f,d^2=i,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efhi,g^a=gi,c^b=ch,d^b=dfhi,g^c=gi,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6560975);